2021 AMC Fall 10B Problem 8

The problem tells us $16384 = 2^{14}$, so $16383 = 2^{14} - 1$, which is a difference of squares:
$$2^{14} - 1 = (2^7 - 1)(2^7 + 1) = 127 \cdot 129.$$
Now $129 = 3 \cdot 43$, and $127$ is prime (it is not divisible by $2, 3, 5, 7$ or $11$, and $11^2 = 121 < 127 < 144 = 12^2$). Thus
$$16383 = 3 \cdot 43 \cdot 127,$$
and the greatest prime divisor is $127$. Its digit sum is $1 + 2 + 7 = 10$.

The answer is $\boxed{\textbf{(C)}\ 10}$.

A number one less than a power of two factors as $2^{2k} - 1 = (2^k - 1)(2^k + 1)$. The first sentence of the problem is not decoration: it points at $2^{14}$ so you can factor algebraically rather than by trial division. Checking primality of $127$ only needs primes up to $11$.

• Trying to factor 16383 by trial division from scratch instead of using the hint 16384 = 2^14 and the difference of squares.
• Stopping at $127 \cdot 129$ and taking $129$ as the largest prime (it is $3 \cdot 43$), or reporting $127$ itself instead of its digit sum.